Properties

Degree 56
Conductor $ 2^{56} \cdot 3^{28} $
Sign $1$
Motivic weight 15
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 1.34e4·4-s − 1.90e6·9-s + 1.24e8·13-s − 2.73e8·16-s + 3.53e11·25-s − 2.57e10·36-s + 3.86e11·37-s + 5.72e13·49-s + 1.67e12·52-s − 1.63e13·61-s − 9.05e11·64-s + 1.58e14·73-s + 3.08e13·81-s − 8.24e14·97-s + 4.77e15·100-s + 2.24e13·109-s − 2.37e14·117-s − 5.78e16·121-s + 127-s + 131-s + 137-s + 139-s + 5.22e14·144-s + 5.20e15·148-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  + 0.411·4-s − 0.132·9-s + 0.550·13-s − 0.255·16-s + 11.5·25-s − 0.0546·36-s + 0.668·37-s + 12.0·49-s + 0.226·52-s − 0.667·61-s − 0.0257·64-s + 1.67·73-s + 0.149·81-s − 1.03·97-s + 4.77·100-s + 0.0117·109-s − 0.0731·117-s − 13.8·121-s + 0.0338·144-s + 0.275·148-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{28}\right)^{s/2} \, \Gamma_{\C}(s)^{28} \, L(s)\cr =\mathstrut & \,\Lambda(16-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{28}\right)^{s/2} \, \Gamma_{\C}(s+15/2)^{28} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(56\)
\( N \)  =  \(2^{56} \cdot 3^{28}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(15\)
character  :  induced by $\chi_{12} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(56,\ 2^{56} \cdot 3^{28} ,\ ( \ : [15/2]^{28} ),\ 1 )$
$L(8)$  $\approx$  $47.9769$
$L(\frac12)$  $\approx$  $47.9769$
$L(\frac{17}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3\}$, \(F_p\) is a polynomial of degree 56. If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 55.
$p$$F_p$
bad2 \( 1 - 1.34e4T^{2} + 4.55e8T^{4} - 8.93e12T^{6} + 2.03e18T^{8} + 4.79e21T^{10} + 1.11e27T^{12} - 2.66e30T^{14} + 1.19e36T^{16} + 5.52e39T^{18} + 2.51e45T^{20} - 1.18e49T^{22} + 6.50e53T^{24}+O(T^{26}) \)
3 \( 1 + 1.90e6T^{2} - 2.72e13T^{4} + 3.17e21T^{6} - 3.98e28T^{8} + 1.04e35T^{10} + 8.33e42T^{12} - 1.24e50T^{14} + 1.71e57T^{16}+O(T^{17}) \)
good5 \( 1 - 3.53e11T^{2} + 6.26e22T^{4} - 7.34e33T^{6} + 6.36e44T^{8} - 4.34e55T^{10}+O(T^{12}) \)
7 \( 1 - 5.72e13T^{2} + 1.60e27T^{4} - 2.97e40T^{6} + 4.12e53T^{8}+O(T^{10}) \)
11 \( 1 + 5.78e16T^{2} + 1.68e33T^{4} + 3.26e49T^{6} + 4.73e65T^{8}+O(T^{9}) \)
13 \( 1 - 1.24e8T + 8.15e17T^{2} - 9.32e25T^{3} + 3.33e35T^{4} - 3.56e43T^{5} + 9.16e52T^{6} - 9.27e60T^{7}+O(T^{8}) \)
17 \( 1 - 4.15e19T^{2} + 8.82e38T^{4} - 1.26e58T^{6}+O(T^{8}) \)
19 \( 1 - 2.28e20T^{2} + 2.67e40T^{4} - 2.13e60T^{6}+O(T^{7}) \)
23 \( 1 + 3.76e21T^{2} + 7.24e42T^{4} + 9.47e63T^{6}+O(T^{7}) \)
29 \( 1 - 1.60e23T^{2} + 1.27e46T^{4} - 6.62e68T^{6}+O(T^{7}) \)
31 \( 1 - 4.80e23T^{2} + 1.14e47T^{4} - 1.78e70T^{6}+O(T^{7}) \)
37 \( 1 - 3.86e11T + 3.80e24T^{2} - 6.10e35T^{3} + 7.31e48T^{4} + 2.36e59T^{5}+O(T^{6}) \)
41 \( 1 - 1.35e25T^{2} + 1.09e50T^{4}+O(T^{6}) \)
43 \( 1 - 3.21e25T^{2} + 5.32e50T^{4}+O(T^{6}) \)
47 \( 1 + 1.55e26T^{2} + 1.20e52T^{4}+O(T^{6}) \)
53 \( 1 - 9.76e26T^{2} + 4.65e53T^{4}+O(T^{6}) \)
59 \( 1 + 7.41e27T^{2} + 2.72e55T^{4}+O(T^{6}) \)
61 \( 1 + 1.63e13T + 1.11e28T^{2} + 1.68e41T^{3} + 6.17e55T^{4} + 8.62e68T^{5}+O(T^{6}) \)
67 \( 1 - 2.13e28T^{2} + 2.65e56T^{4}+O(T^{6}) \)
71 \( 1 + 8.14e28T^{2} + 3.23e57T^{4}+O(T^{6}) \)
73 \( 1 - 1.58e14T + 1.22e29T^{2} - 1.74e43T^{3} + 7.80e57T^{4} - 1.03e72T^{5}+O(T^{6}) \)
79 \( 1 - 4.09e29T^{2} + 8.13e58T^{4}+O(T^{6}) \)
83 \( 1 + 9.72e29T^{2} + 4.72e59T^{4}+O(T^{5}) \)
89 \( 1 - 3.65e30T^{2} + 6.53e60T^{4}+O(T^{5}) \)
97 \( 1 + 8.24e14T + 1.02e31T^{2} + 8.41e45T^{3} + 5.35e61T^{4}+O(T^{5}) \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{56} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−1.94803837266857507645134763871, −1.78525848636299174880348817249, −1.76674585203865871646199227512, −1.54290340883853369675145667251, −1.53121098929053550213570698543, −1.37505467498819744467892916753, −1.36116287774433329404231250940, −1.30318396620962685946705573053, −1.23200562899961258919165597775, −1.17632104744704135824400057571, −1.14869408976486925641633082849, −1.06346101308968340095286253043, −0.997045603762658054585504930147, −0.898348320189514778052369284822, −0.890608848914073938432205962272, −0.870567588653024221607501967607, −0.74829149169094346308338404316, −0.56930173951345074840501258625, −0.56659344872335834166221963959, −0.55719007597664492381891425793, −0.46910028136008476455112447387, −0.31503957298744116983677893217, −0.26761294510943663854831188483, −0.097357163171033544571664301198, −0.080918165437846444868058449151, 0.080918165437846444868058449151, 0.097357163171033544571664301198, 0.26761294510943663854831188483, 0.31503957298744116983677893217, 0.46910028136008476455112447387, 0.55719007597664492381891425793, 0.56659344872335834166221963959, 0.56930173951345074840501258625, 0.74829149169094346308338404316, 0.870567588653024221607501967607, 0.890608848914073938432205962272, 0.898348320189514778052369284822, 0.997045603762658054585504930147, 1.06346101308968340095286253043, 1.14869408976486925641633082849, 1.17632104744704135824400057571, 1.23200562899961258919165597775, 1.30318396620962685946705573053, 1.36116287774433329404231250940, 1.37505467498819744467892916753, 1.53121098929053550213570698543, 1.54290340883853369675145667251, 1.76674585203865871646199227512, 1.78525848636299174880348817249, 1.94803837266857507645134763871

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.